While the first term in (REF ) is easily approximated given any dataset, the second term requires sampling the induced model distribution, which is known to be hard in high dimensions [1]}.
For any reasonably sized image \( x \in \mathbb {R}^n \)  computing the partition function is infeasible, hence the distribution has to be approximated using mcmc techniques.
In this work, we utilize the ula [2]}, [3]}, [4]}, which makes use of the gradient of the underlying probability density function to improve mixing times of the Markov chains.
The ula algorithm read as
\(x^k \sim \mathcal {N}(x^{k - 1} + \frac{\epsilon }{2}\nabla _1\log p_\phi (x^{k - 1}), \beta \epsilon \mathrm {Id}_n),\ k = 1,\cdots ,K,\) 
